Geant4 Model for the Stopping Power of Low Energy Negatively Charged Hadrons St´Ephane Chauvie, Petteri Nieminen and Maria Grazia Pia

1 Geant4 model for the stopping power of low energy negatively charged hadrons St´ephane Chauvie, Petteri Nieminen and Maria Grazia Pia

Abstract— An original model is presented for the simulation measurements; results from more sophisticated calculations of the energy loss of negatively charged hadrons. It calculates are also discussed: the agreement with both experimental data the stopping power by regarding the target atoms as an en- and state-of-the art theoretical calculations demonstrate the semble of quantum harmonic oscillators; this approach allows to account for charge dependent effects in the stopping power, validity of the method proposed and its appropriateness for which are relevant at energies below a few MeV. The resulting Monte Carlo computation. antiproton stopping powers for different elements are shown This model is implemented in the Geant4 [1], [2] Low En- to be in satisfactory agreement with experimental data. The ergy Electromagnetic package [3], [4]; it is publicly available model described in this paper is implemented in the Low Energy as part of the Geant4 toolkit. Electromagnetic package of the Geant4 Toolkit; it represents a significant improvement for the accurate simulation of low energy Various experiments, like antimatter studies in space exper- negative hadrons with respect to previously available models. iments [5] or the recent interest for biological applications of low energy antiprotons [6], [7], may profit of the specialized Index Terms— Monte Carlo, Geant4, simulation, stopping power. modelling approach presented in this paper, together with the advanced functionality and modern software technology offered by Geant4 as a general purpose Monte Carlo system. I. INTRODUCTION

HE stopping power of matter for charged particles is an II. FUNDAMENTAL CONCEPTS OF THE STOPPING POWER T important physics quantity in various applications. Its ac- OF HEAVY CHARGED PARTICLES curate knowledge over a broad range of energies and materials for both positively and negatively charged particles is essential The stopping power of a point charge penetrating through to a variety of ongoing and future physics experiments. matter at non-relativistic speed is conventionally described by Bethe’s theory [8], [9], which is based on two main assump- A reliable calculation of the stopping power of charged particles with kinetic energies below a few MeV/u is required tions: the stopping power is caused by Coulomb excitation and in Monte Carlo simulation for the precise description of energy ionization of the electrons in the stopping medium, and the interaction can be treated within the first Born approximation. loss processes and the correct estimate of their ranges in matter. Bethe’s theory describes inelastic collisions in which the target atoms or molecules are excited or ionized. The nuclear This paper describes the development of an original model stopping power, arising from the energy transfer to atoms that for the precise simulation of the electromagnetic interactions recoil as a whole, becomes important at energies below several of negative hadrons at energies around and below the Bragg tens of keV/u; it will not be considered in this work. peak, based on the quantal harmonic oscillator approach; it For a projectile of charge Z, mass M and velocity v (kinetic proposes a systematic procedure to determine the oscillator 1 2 energy E = 2 Mv ), which traverses a medium of atomic strengths and resonance energies, which constitute the basic Z N parameters of the underlying theoretical model. The resulting number m with m target atoms per unit volume, the energy method combines simplicity and accuracy; these characteris- loss per unit path length (i.e. the stopping power) is given by [8], [9] tics make it suitable for application in Monte Carlo simulation. 4 dE 4πe 2 It is worthwhile reminding the reader that the lack of system- S ≡− = Nm Zm Z L0, dx m v2 (1) atic and precise experimental measurements of the antiproton e stopping power over the whole periodic system of elements where me and −e are the electron mass and charge, respec- prevents a simulation approach based on the parameterisation tively. The dimensionless stopping number L 0 is defined as of experimental data. Z 3 “ ” mev X d q 2 2 The stopping powers for antiprotons obtained with the L0 = (En − E0) δ En − E0 − q·v + q /2M |Fn0 (q)| , (2) π n q4 simulation model described are compared to experimental where En is the energy of the target state |n, obtained by Manuscript received January 12, 2007. solving the Schr¨odinger equation H|n = En|n, with the S. Chauvie is with Azienda Ospedaliera Santa Croce e Carle Cuneo and Hamiltonian H describing one target atom or molecule. In INFN Sezione di Torino, I-10125, Torino - Italy (phone: +39 0171 641558, q fax: + 39 0171 641564; e-mail:[email protected]). equation (2) is the momentum transfer in the collision; P. Nieminen is with ESA-ESTEC, Keplerplan 1, 2200 AG Noordwjik - The      Zm  Netherlands (e-mail: [email protected]).    F (q)= n exp(iq·r /¯h)0 M.G. Pia is with INFN Sezione di Genova, Via Dodecaneso 33, I-16146 n0  j  (3) Genova - Italy (e-mail: [email protected]). j=1 2

is the so-called inelastic form factor, closely related to the gen- G4hLowEnergyIonisation eralized oscillator strength [9], which in turn is proportional to the conditional probability that the target makes a transition |0 |n from the ground state to a particular excited state upon 1..n receiving a momentum q; rj are the coordinates of the atomic electrons. G4VLowEnergyModel If the ground state of the target is isotropic, (2) may be rewritten as   1 dQ 2 L0 = (En − E0) |Fn0(q)| , (4) 2 Q2 G4FreeElectronGasModel G4hBetheBlochModel n where the expression within angular brackets indicates an G4QAOLowEnergyLoss angular average, and the integration limits are obtained from  m 2 E − E + e Q ≤ 2m v2Q n 0 M e (5) Fig. 1. Class diagram of the main design features concerning negatively charged hadrons in the Geant4 Low Energy Electromagnetic package. 2 with the recoil energy defined by Q ≡ q /2me. In the case of projectile speed v much greater than the orbital speed of the target electrons, one can circumvent the III. THE ENERGY LOSS OF NEGATIVELY-CHARGED evaluation of the inelastic form factor. In this high-energy HADRONS IN GEANT4 limit, the asymptotic expression reads The Geant4 toolkit is a general-purpose Monte Carlo code

2 2meM v for the simulation of the passage of particles through matter. L0  LBethe ≡ ln , (6) Different Geant4 domains are responsible for the geometry me + M I definition, primary particle generation, simulation of particle which is the familiar Bethe formula; I is the mean excitation transport, visualization, and for other aspects of the Monte energy [10]. In general, however, one has to calculate the Carlo simulation; in particular, the Low Energy Electromag- Fn0(q) factor. The analytical evaluation of Fn0(q) is feasible netic Package offers various models for electrons and photons only for very simple systems, namely the hydrogen atom and [13], hadrons [14] and ions [15], with special care to low- the free-electron gas. For other systems, the inelastic form energy interactions and the accurate definition of energy factor should be obtained numerically. losses. Formally one may generalize the stopping power formula Thanks to the object oriented technology adopted and (1) including higher-order terms in the stopping number, 2 its sound design, the Geant4 Low Energy Electromagnetic replacing Z L0 by package provides a variety of physics models to describe 2 2 3 4 the stopping power of hadrons. The different models are Z L = Z L0 + Z L1 + Z L2 + ... (7) specialized according to the energy range and particle type; This expression emerges from a Born expansion of the stop- they are handled transparently by the class responsible for the ping power. The first term in the right-hand side represents hadron ionisation process G4hLowEnergyIonisation, as they the first Born approximation already described, the second is all obey the same abstract interface G4VLowEnergyModel. the Barkas (or Z 3) correction [11] and the third is the Z 4 Complementary models appropriate different energy ranges correction, part of which included in Bloch’s scheme [12]. can be composed to cover an extended energy interval in a It is worthwhile noticing that the presence of terms with simulation application; the user has the possibility to select odd powers in Z in (7) leads to a different stopping behaviour different models as appropriate to the experimental applica- of positively and negatively charged particles. This effect in- tion. creases with decreasing velocity of the projectile, and reaches In the case of negatively charged hadrons, three modelling a maximum when this is comparable to the velocity of the approaches are adopted in the Low Energy Electromagnetic electrons in the medium. package to describe the energy loss with adequate simulation It is important to stress that the Born-series expansion accuracy at different energies of the interacting particle. The (7) does not account for other effects in the stopping of class diagram in Fig. 1 shows the main design features heavy particles, such as, for instance, the capture and loss concerning this domain; the diagram is expressed in UML of electrons from the medium in the course of the slowing (Unified Modeling Language) [16]. down of positively-charged particles. This phenomenon is The Barkas effect plays a significant role in the energy often treated in the various formalisms by assigning to the loss of hadrons at intermediate and low energies, but it can “dressed” projectile a velocity-dependent effective charge. be neglected at relatively high energies; therefore the same In this respect, it has been said that antiprotons are “the Bethe–Bloch model for the energy loss of both positive and theorist’s favourite low-energy projectile”: the lack of bound negative hadrons is implemented in Geant4 for kinetic energies electron-antiproton states makes them travel as bare, undressed E>E2 = 2 MeV/u. In the Low Energy Electromagnetic particles, thus simplifying the theoretical description of the package implementation the the Bethe formula is supple- inelastic interactions experienced by them. mented with shell corrections and relativistic effects, including 3 the density-effect correction [14]. for instance [29]); this is also the case for the higher-order L 1 The extended oscillator model described in the follow- and L2 terms. Alternative approaches, such as the dielectric ing section has been specifically developed for the Geant4 formalism (local-density approximation) [20] or the kinetic Low Energy Electromagnetic package to handle projectile theory [30], share similar difficulties; therefore they are not kinetic energies E1 quantum mechanics; these an increase of the antiproton stopping power with decreasing authors were able to derive an expression for the stopping L velocity below ∼ 0.5–0.7 keV/u [22], [23]; this behaviour number 0, that they evaluated by numerical integration [17]. might be attributed to the sizeable contribution of nuclear The stopping number was shown to approach the asymptotic stopping, but the issue is still under debate [24], [25]. Bethe logarithm at high speed (6) and to correctly reproduce Since typical hadron masses are much larger than the the leading term in the shell corrections of both the stopping electron mass, the ionization energy loss depends only on and straggling cross sections; calculations performed for the the hadron velocity and on charge [26]–[28]; therefore the hydrogen atom yielded encouraging results. Within a few stopping power of a charged hadrons with mass M, Z = −1 years, the same group of researchers succeeded in obtaining L L and kinetic energy E is equal to that of an antiproton (mass expressions for the Barkas 1 and Bloch 2 terms correspond- ing to the quantal harmonic oscillator [18], [19]; their results mp) with the same velocity; the corresponding kinetic energy of the antiproton is were again given in the form of tables. Unfortunately, the mp calculation of further terms in the sum (7) seems prohibitive Ep = E . (9) M due to the mathematical complexity already encountered in the 4 This scaling can be used for particles with |Z| =1only. terms O(Z ). Hence the following sections will discuss only the simulation The obvious next step is to extend the results found for of antiproton energy loss, keeping in mind that the stopping a harmonic oscillator to the description of real atoms. To power of other heavy particles with Z = −1 at any energy this end, resonance frequencies and oscillator strengths have can be derived from the antiproton ones. to be assigned to relevant electronic transitions. As a crude model, it could be possible to describe the whole atom as a IV. GEANT4 MODEL FOR LOW ENERGY ANTIPROTON single oscillator with the resonance energy set equal to I, the STOPPING POWER mean excitation energy: this constitutes a good approximation This paper addresses in detail the development of a stopping for hydrogen and helium [34], but for heavier elements the power model specialized for low energy negatively charged contributions of the different atomic shells should be evaluated hadrons, suitable for application in the Geant4 toolkit. In separately. Consequently, Sigmund and co-workers [17], [34] proposed to determine the stopping number of an atom from this context simple, still physically motivated models for the  2 calculation of the stopping number are particularly appropriate 2mev L = f L . for simulation purposes. An important requirement for the atom n ¯hω (10) n n implementation in a general-purpose Monte Carlo system is also the wide computational applicability of a theoretical Here, ωn ≡ (En − E0)/¯h are resonance frequencies and   method: in fact, a key issue for a physics model to be used in    2   Zm   the software is the possibility of its calculation for a variety 2me     fn = ¯hωn n rj 0 (11) of interacting material likely to be encountered in the course 3¯h2      j=1  of a simulation. The numerical evaluation of the inelastic form factor (3) in are the associated optical (dipole) oscillator strengths; these the stopping number L0, equation (2), is a complex issue (see are actually distributions df / d (¯hω), since most excited atomic 4 states lie in the continuum, and the sum in (10) should be the inclusion of the so-called Lorentz–Lorenz correction. In understood as an integral over ¯hω. This extended oscillator metals, the effective number of conduction electrons f n is set model expresses the well-known fact that an atom may to equal to the lowest chemical valence of the considered element a certain extent be regarded as an assembly of harmonic and ¯hωn = fn/Zm ¯hωp. The semi-empirical adjustment oscillators. Similar decompositions of the stopping power factor a is introduced to ensure agreement with the adopted S(E) into contributions from separate shells are performed mean excitation energy I through equation (13). The physical as well in the kinetic theory of stopping [40]. meaning of a is made clear if one recalls that the contribution In order to reproduce the asymptotic Bethe formula (6), the of the ionization of a given inner shell n to continuum states excitation energies and oscillator strengths must satisfy the involves energies which are larger than the corresponding ab- Thomas–Reiche–Kuhn sum rule (i.e. the Bethe sum rule for sorption edge Un. In the calculations reported below Un and I Q =0) [9]  have been taken from [43] and [10] respectively. It is expected fn = Zm (12) that a ∼ exp(1/2) ≈ 1.65, at least for inner shells [41]; the n actual values of a generally lie in the range 1.5–2.5. Chemical and lead to the accepted value of the mean excitation energy phase effects may be incorporated in a straightforward manner I through the relation [9] by selecting a suitable for the considered material.  The method presented here is simple and enables a system- fn ln(¯hωn)=Zm ln I. (13) atic evaluation of the parameters ωn and fn, which enter the n expression of the stopping power within the framework of the Instead of using the detailed excitation/ionization spectrum quantal harmonic oscillator model. These characteristics make of the atom, it is more advantageous to adopt in (10) a discrete it suitable for the implementation in a Monte Carlo system for set of oscillators, usually one frequency and oscillator strength particle transport. per atomic bound shell, constrained to fulfill equations (12) and (13). These parameters are then obtained by means of B. Implementation details shell-wise integration of available (experimental or theoretical) oscillator strength data in the form of optical or dielectric Antiproton stopping powers have been calculated employing response functions. This is nevertheless a laborious procedure, the quantal harmonic oscillator model with parameters f n and ω and has only been carried out for a few selected materials, n determined from equations (14) and (15). For antiprotons Z = −1 L = L − L + L mainly metals; tabulated values of ωn and fn for some ( ) one has 0 1 2; the different elements can be found in [25]. contributions are evaluated by means of a linear interpolation L The outlined method for the determination of the ω n and in the original tables of i values presented in [17]–[19]. The fn parameters is still too complex to be of practical use in various stopping number contributions as a function of the a general-purpose Monte Carlo code; further simplifications projectile kinetic energy are shown in Fig. 2 for antiprotons are thus necessary. Here the approach used in [41], [42] is interactions with silicon; the energy range from 5 keV to 2 followed to compute the density-effect correction for electrons MeV is considered. and positrons, that is the reduction of the stopping power of As expected, the largest contribution to L comes from the fast charged particles due to the dielectric polarization of the first-order term L0. The (first) Barkas term L1 is relevant up medium. Firstly, the oscillator strength is simply set equal to to energies ∼ 500 keV, especially around the Bragg peak, the number of electrons in the n-th atomic shell, that is which is located at ∼ 100–200 keV; it contributes by roughly 30% to the total stopping power. The L2 (first Bloch) term fn = Zn. (14) is always much smaller than the others and becomes negative above approximately 100 keV. The Thomas–Reiche–Kuhn sum rule (12) is automatically S(E) fulfilled with this prescription. It is worth mentioning that this The calculated curves for antiproton stopping in choice implies neglecting the transfer of oscillator strength aluminium, silicon, and gold are displayed in Fig. 3-5. These elements span a wide range of atomic numbers, and are from inner to outer shells, but this approximation has a very Z limited impact on calculated stopping powers, as shown in representative of lower and higher m. The stopping powers [40] for the kinetic theory of particle stopping. Secondly, the of protons in the same materials, as tabulated in [44], are also resonance frequencies can be obtained from shown for comparison purposes. The difference between the proton and antiproton stopping powers is primarily due to the 2 f 2 n 2 Barkas effect and, to a lesser extent, to the aforementioned ¯hωn = (aUn) + (¯hωp) , (15) 3 Zm electron capture and loss processes for the positive projectiles. At energies above ∼ 1 MeV the S(E) curves for both particles where Un is the ionization energy of the n-th shell and the tend to merge, and eventually the stopping power is accurately nominal plasmon energy is defined by predicted by the Bethe–Bloch formula. 2 2 ¯hωp ≡ 4π¯h e NmZm/me. (16) This procedure accounts for the well-known fact that in a C. Validation of the model dense medium the atomic frequencies are displaced to higher The predictions of the extended harmonic oscillator model values [33]. The factor 2/3 in equation (15) arises from implemented in Geant4 have been compared to experimental 5

The situation is slightly worse for gold (Fig. 5), where discrepancies of the order of 20–30% are found. For these elements the measured values seem to indicate a flatter Bragg peak than the one obtained from the calculations. However, the large spread of experimental data does not permit the extraction of definitive conclusions until further experiments are undertaken. It is nevertheless clear that the use of proton stopping powers considerably overpredicts the data for antiprotons. In summary, from the present comparison the global agreement between the measured stopping powers and the quantal har- monic oscillator model with parameters fn and ωn determined as described in this paper looks satisfactory; it represents a significant improvement of the achievable simulation relia- bility with respect to the model for protons, which was the only approach for hadron ionisation available in the Geant4 Low Energy Electromagnetic package prior to the present development. For antiproton kinetic energies below around 30 keV only limited experimental data are available [21]. The scatter of the Fig. 2. Stopping number for antiprotons in Silicon as resulting from the sum of the terms in Born expansion: L0 (Bethe term), L1 (Barkas term), L02 measured values is rather large, and is caused by fluctuations (Bloch term),and the total stopping number L; the vertical scale is in arbitrary in the intensity of the pulsed antiproton beam. However, the units. effect of a stopping power proportional to the velocity of the projectile is clearly observable for gold and well reproduced by the present model. data collected with the LEAR antiproton beam at CERN [45]– In the case of aluminium and silicon, more sophisticated [49]. This paper presents a first comparison with selected data calculations of antiproton stopping power have been performed sets of relevant experiments for the evaluation of the soundness by Arista and Lifschitz [31], [32]. These authors employed a of the modelling approach; a more comprehensive validation self-consistent method based on the extension of the Friedel study of the Geant4 model developed is foreseen as the object sum rule to finite velocities, which incorporates non-linear of a dedicated future paper. effects in the description of the energy loss of heavy charged The experiments concerned were carried out by first degrad- particles. The resuls of this approach are in similar agreement ing the 5.9 MeV antiprotons from the accelerator to the desired with the cited experimental values as the extended harmonic kinetic energy, then measuring the change in their velocity, by oscillator model presented in this paper; this demonstrates that means of a time-of-flight technique, on passing through foils the simple approach adopted for the Geant4 model is anyway of various compositions and thicknesses. adequate for Monte Carlo simulation applications. The measured stopping powers for aluminium, silicon, and gold are plotted in Fig. 3 to 5. These experimental values show significant variations, reflecting the difficulties inherent V. C ONCLUSION to such measurements. In particular, systematic errors arise A specialized model for the energy loss of negative hadrons from the imprecision in the determination of the thicknesses of has been developed for the Geant4 Low Energy Electromag- the foils used as stopping material, which are in the range 0.31 netic package. It is based on the quantal harmonic oscillator to 6.9 μm, and from the finite resolution of the time-of-flight model, which provides a simple, yet sufficiently accurate method [48], [49]. For instance, the antiproton data of [47] theoretical framework for the calculation of stopping powers had to be renormalized [48], [49] because the gold foils were suitable for Monte Carlo transport applications; it is applicable found to be 17% thinner than specified by the manufacturer. to particles of energy between 25 keV/u and 2 MeV/u. One should bear these limitations in mind when comparing The agreement between this model and available measure- the experimental results with theoretical calculations. ments is satisfactory for simulation applications. In the case For aluminium and silicon (Fig. 3 and 4) the agreement of low-atomic-number elements, the differences do not exceed between theory and experiment above several tens of keV is ∼ 10%, whereas for elements with high atomic numbers quite good, with differences smaller than ∼ 10%. For these they can reach up to ∼ 30%. Nonetheless, even in this case two elements the measured values cover a wide energy range the calculation based on the model developed represents an and yield a well-defined maximum of S(E). Around 200 keV, improvement with respect to the previously available simu- the Geant4 calculation for silicon is in better agreement with lation tools, since the use of proton stopping powers in a the data reported in reference [48], [49] than with the older simulation would lead to errors around 50–60%. The data measurements [45], [46]. The latter values are somewhat from the extended harmonic oscillator model are in agreement lower, presumably due to a systematic error in the energy with other theoretical calculations based on more sophisticated calibration [48], [49]. models. 6

Fig. 3. Stopping powers for antiprotons in aluminium: the solid curve is Fig. 4. Stopping powers for antiprotons in silicon: the solid curve is the result the result of the Geant4 model; the dashed curve is the stopping power for of the Geant4 model; the dashed curve is the stopping power for protons [44]; protons [44]; the dots are experimental data from [21], [48], [49]. the dots are experimental data from [45], [46], [48], [49]; the dotted curve is the result of the more sophisticated theoretical model in [31], [32].

At the present time, the development described in this paper is the only dedicated model for the precise simulation of [10] “Stopping Powers for Electrons and Positrons”, ICRU Report 37, ICRU, antiproton stopping power publicly released in general-purpose Bethesda, MD, 1984. [11] W. H. Barkas, J. N. Dyer and H. H. Heckman, “Resolution of the Σ− Monte Carlo codes handling hadrons, like FLUKA [50]–[52], Mass Anomaly” Phys. Rev. Lett., vol. 11, pp 26-28, 1963 Erratum: ibid. Geant4 and MCNP versions (MCNP5 [53], [54] and MCNPX pp. 138. [55], [56]). [12] F. Bloch, “Zur Bremsung rasch bewegter Teilchen beim Durchgang durch die Materie”, Ann. Phys. (Leipzig), vol. 16 pp 285-320, 1933. [13] J. Apostolakis, S. Giani, M. Maire, P. Nieminen, M.G. Pia, and L. Urban, ACKNOWLEDGMENT “Geant4 low energy electromagnetic models for electrons and photons” INFN/AE-99/18, Frascati, 1999. We thank Prof. N. R. Arista (Centro At´omico Bariloche, [14] S. Giani, V. N. Ivanchenko, G. Mancinelli, P. Nieminen, M. G. Pia, Argentina) and Prof. S. P. Møller (University of Aarhus, Den- and L. Urban, “Geant4 simulation of energy losses of slow hadrons”, mark) for providing us numerical data of antiproton stopping INFN/AE-99/20, Frascati, 1999. [15] S. Giani, V. N. Ivanchenko, G. Mancinelli, P. Nieminen, M. G. Pia, and powers, Prof. J. M. Fernandez Varea (University of Barcelona, L. Urban, “Geant4 simulation of energy losses of ions”, INFN/AE-99/21, Spain) for his valuable theoretical support, and Prof. V. N. Frascati, 1999. Ivanchenko (Budker Institute, Novosibirsk, Russia) for fruitful [16] G. Booch, J. Rumbaugh, and I. Jacobson, “The Unified Modeling Language User Guide”, Ed. Boston: Addison-Wesley, 1999. discussions in the early phase of the software implementation. [17] P. Sigmund and U. Haagerup, “Bethe stopping theory for a harmonic oscillator and Bohrs oscillator model of atomic stopping”, Phys. Rev. A, vol. 34, pp. 892-910, 1986, Erratum: ibid., vol. 35, pp. 3965, 1987. REFERENCES 4 [18] H. H. Mikkelsen, “The Z1 -term in electronic stopping: impact parameter [1] S. Agostinelli et al., “Geant4 - a simulation toolkit”, Nucl. Instrum. Met. dependence and stopping cross section for a quantal harmonic oscilla- A, vol. 506, no. 3, pp. 250-303, 2003. tor”, Nucl. Instrum. Meth. B, vol. 58, pp. 136-148, 1991 [2] J. Allison et al., “Geant4 Developments and Applications”, IEEE Trans. [19] H. H. Mikkelsen and P. Sigmund, “Impact parameter dependence of Nucl. Sci., vol. 53, no. 1, pp. 270-278, 2006. electronic energy loss: Oscillator model” Nucl. Instrum. Meth. B, vol. [3] S. Chauvie, G. Depaola, V. Ivancheko, F. Longo, P. Nieminen, and M. G. 27, pp. 266-275, 1987. Pia, “Geant4 Low Energy Electromagnetic Physics”. in Proc. Computing [20] J. Lindhard and A. Winther, “Stopping power of electron gas and in High Energy and Nuclear Physics, Beijing, China, pp. 337-340, 2001. equipartition rule”, K. Dan. Vid. Selsk. Mat.-Fys. Medd., vol. 34, no. [4] S. Chauvie et al., “Geant4 Low Energy Electromagnetic Physics”, in 4, 1964. Conf. Rec. 2004 IEEE Nucl. Sci. Symp., N33-165. [21] S. P. Møller, A. Csete, T. Ichioka, H. Knudsen, U. I. Uggerhøj and H. [5] B. Borgia, “The Alpha Magnetic Spectrometer on the International H. Andersen, “Antiproton Stopping at Low Energies: Confirmation of Space Station”, IEEE Trans. Nucl. Sci., vol. 52, no. 6, pp. 2786-2792 , Velocity-Proportional Stopping Power”, Phys. Rev. Lett., vol. 88, pp. 2004. 193201-5, 2002. [6] C. Maggiore et al., “Biological effectiveness of antiproton annihilation”, [22] A. Adamo et al., “Antiproton stopping power in hydrogen below 120 Nucl. Instr. Meth. B, vol. 214, pp. 181-185, 2004. keV and the Barkas effect”, Phys. Rev. A, vol. 47, pp. 4517-4520, 1993. [7] M. H. Hozscheiter et al., “Biological effectiveness of antiproton annihi- [23] M. Agnello et al., “Antiproton Slowing Down in H2 and He and lation”, Nucl. Instr. Meth. B, vol. 221, pp. 210-214, 2004. Evidence of Nuclear Stopping Power”, Phys. Rev. Lett., vol. 74, pp. [8] H. Bethe, “Zur Theorie des Durchgangs schneller Korpuskularstrahlen 371-374, 1995. durch Materie” Ann. Phys., (Leipzig) vol. 5, pp. 325-400, 1930. [24] G. Schiwietz, U. Wille, R. D´ıez Mui˜no, P.D. Fainstein and P.L. Grande, [9] M. Inokuti, “Inelastic Collisons of fast Charged Particles with Atoms “Comprehensive Analysis of the Stopping Power of Antiprotons and and Molecules - the Bethe Theory Revisited” Rev. Mod. Phys., vol. 43, Negative Muons in He and H2 Gas Targets”, J. Phys. B: At. Mol. Opt pp. 297-347, 1971. Phys., vol. 29, pp 307-321, 1996. 7

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